Multiplying polynomials without losing a term
Every term in one factor multiplies every term in the other. Organize the products, then combine like terms once.
How to recognize the method, run it, and know when it is the wrong choice.
Updated July 13, 2026The distributive property—not a special binomial trick—creates every product term.
Distribution scales to any number of terms
For each term in the first polynomial, distribute across the entire second polynomial. A grid or vertical layout can make the pairings visible.
FOIL is only the two-by-two case. Relying on the acronym becomes fragile as soon as a trinomial appears.
Multiply coefficients and add exponents
Within each product, multiply the numerical coefficients and use the product rule for matching variable bases.
Carry signs with their coefficients. Writing negative terms explicitly reduces accidental sign changes.
Combine only after all products exist
Premature combining makes it easy to omit a pairing. List the products first, then group identical powers.
A degree check helps: the product degree should usually equal the sum of factor degrees when leading coefficients are nonzero.
Expand (2x − 3)(x + 5).
Distribute each term in the first binomial.
Multiply each pair.
Combine the like x-terms.
Common mistakes
- Multiplying only the first and last terms.
- Adding rather than multiplying coefficients.
- Combining unlike powers of x.
Three takeaways
- Every term pairs with every term.
- Signs belong to coefficients.
- Combine like terms at the end.